You can go to Bhutan via flight to paro if it's not available then you can try alternative path -- Delhi to Bagdogara flight which is in Siliguri and then take cab or bus to phunsolling check post in Bhutan.
Hi Shruti, There seems to be some issue with the websites that you have mentioned above. Maybe there's a bug and you will be able to see fresh flights in a day or two. Meanwhile, you can check out flights available on Skyscanner.com. Here's the link: http://bit.ly/2KdOkHZ All flights to Bhutan from Delhi make a stop at Baghdogra, and then one has to change carriers. Only Druk Airways and Bhutan Airlines fly to Paro. Flights are available on Bhutan Airlines' website. Check out flights here: http://www.bhutanairlines.bt/ Hope this has been of help :) - ET